Grant: Discovery Grants Program - Individual

My research is focused on enhancing our understandings of and improving methods used in the analysis correlated data. This proposal particularly focuses on multivariate linear and bilinear models. The generalized multivariate analysis of variance (GMANOVA) model is a multivariate bilinear model that arises when linear restrictions are imposed on the mean parameters of the multivariate analysis of variance (MANOVA) model. Unlike MANOVA, GMANOVA allows a structured mean, where the functional form of the mean trajectory (eg. over time) is incorporated in the model. The estimated mean trajectory for GMANOVA involves bilinear projections, which leads to a design space that can be represented as the tensor product of the column spaces of two design matrices. One of the projections involves weighted inner products, with the inverse of the sample covariance matrix being the most used weight. The use of this random weight limits our understandings of the sampling distributions of the estimators & tests. Moreover, the benefit of weighting & the choice of weights have yet to be explored. This is particularly important considering that the unknown covariance matrix is not optimally estimated in some situations (eg. small sample, singularity). Inference for multivariate linear and bilinear models is traditionally done under the assumption of multivariate normality. We recently provided robust estimators by using the multivariate skew normal distribution. More needs to be done to allow optimal analysis of outcomes with heavy tailed distributions. My research program has a long-term goal of developing novel and improved statistical methods useful in the analysis of correlated outcomes with a focus on multivariate linear and bilinear models useful in the analysis of repeated measurements, longitudinal & clustered longitudinal data. The overarching vision is to maximize optimality and ensure robustness of statistical inference, the including flexibility to allow data integration and provide high-dimensional extensions. Recent progress from my previous NSERC discovery grant and the methodological gaps (limitations) identified in my collaborative research motivated and informed my specific short-term objectives: a) Iteratively Re-weighted Inference: We will study unweighted vs weighted methods and quantify relative optimality; develop a novel weighting strategy; and provide a statistical and computational algorithms (procedures) for recursively re-weighted inference. b) Tests useful for clustered longitudinal data: We will derive tests for the general linear hypothesis for the extended GMANOVA with multivariate normal errors. c) Robust tests for multivariate linear and bilinear models: We will relax the assumption of normality and develop tests under multivariate skewed normal distribution. d) Inference under heavy tail distributions: We will use the multivariate t and skewed t distributions for optimal analysis of outcomes with heavy tailed distributions.